Elucidation of the mechanism of subunit exchange in αB crystallin oligomers

AlphaB crystallin (αB-crystallin) is a key protein for maintaining the long-term transparency of the eye lens. In the eye lens, αB-crystallin is a “dynamical” oligomer regulated by subunit exchange between the oligomers. To elucidate the unsettled mechanism of subunit exchange in αB-crystallin oligomers, the study was carried out at two different protein concentrations, 28.5 mg/mL (dense sample) and 0.45 mg/mL (dilute sample), through inverse contrast matching small-angle neutron scattering. Interestingly, the exchange rate of the dense sample was the same as that of the dilute sample. From analytical ultracentrifuge measurements, the coexistence of small molecular weight components and oligomers was detected, regardless of the protein concentration. The model proposed that subunit exchange could proceed through the assistance of monomers and other small oligomers; the key mechanism is attaching/detaching monomers and other small oligomers to/from oligomers. Moreover, this model successfully reproduced the experimental results for both dense and dilute solutions. It is concluded that the monomer and other small oligomers attaching/detaching mainly regulates the subunit exchange in αB-crystallin oligomer.


S2
The scattering profiles of OLG(26) at the concentration of 0.45 mg/mL in full D 2 O buffer

Exchangeable subunit number in oligomer
We adopted the average association number of αB-crystallin oligomers as 26. Here, we estimated the number of exchangeable subunits n in the αB-crystallin oligomer with an association number of 26. The A value in eq. (2), which is the ratio of intensity at the equilibrium state to the initial intensity, is a function of n, and then A(n) is expressed by the following equation:

Collision model
We assumed that subunit exchange could occur due to a random collision between two αBcrystallin oligomers. It is also assumed that the αB-crystallin oligomers always exchange their subunits when the two oligomers collide. Namely, the collision frequency calculated from this model is comparable to the exchange rate. I 0 (t) was calculated using the following procedure.
1. We set the parameters and a function as follows: • Collision frequency between two oligomers: k. (k l at dilute sample, k h at dense sample) • Forward scattering intensity at cs: I 0 (cs) • Normalized forward scattering intensity at cs: I 0, nor (cs) = I 0 (cs)/I 0 (0) 2. Supposing the collision between OLG(i) and OLG(j) at cs, the numbers of six oligomers, OLG(i), OLG(i-1), OLG(i+1), OLG(j), OLG(j-1), and OLG(j+1), will be affected at cs+1. We should also consider that there are two cases of subunit exchange. One is that the pd-subunit in OLG(i) exchanges the h-subunit in OLG(j) (Case 2-1). Because both the number of pd-subunits in OLG(i) and that of the h-subunit in OLG(j) is proportional to the collision probability, the collision probability is given by (i/26) * ((26 -j)/26) in Case 2-1. The other is that the h-subunit in OLG(i) exchanges with the pdsubunit in OLG(j) (Case 2-2). In Case 2-2, the collision probability is given by (26-i)/26) * (j/26).
Depending on the cases (Case 2-1 and Case 2-2), the following equations as a function of cs can be written. that can reproduce I 0 (t) of the dense sample was estimated. The results are plotted in Fig. S6 (b).
7. Finally, the collision ratio for the dense sample to the dilute sample (= k h /k d ) was calculated to be 31.5. This figure is prepared by the usage of IGOR Pro 6.34A (https://www.wavemetrics.com/forum/news-andannouncements/igor-634a-now-shipping). S10 8. It is considered that the collision frequency is dependent on both the diffusion constant and the concentration of particles. We can calculate the collision frequency from another experimental approach for the examination of the collision frequency rate obtained by iCM-SANS under the collision model. Based on the idea of a diffusion-limited reaction, the collision frequency, k ex , is given by the following equation: where N j , D j , and R j correspond to the number density, translational diffusion constant, and collision radius of j-particle (j = a, b), respectively. Assuming that the collision radius is equal to the hydrodynamic radii,

Monomer Attaching/Detaching model
It is considered that attaching/detaching monomers contribute to the subunit exchange in αBcrystallin oligomers (refer to Fig. 5). I 0 (t) was calculated using the following procedure.
1. We defined the rates of attaching and detaching monomers as k a and k d , respectively.
2. In order to consider the distribution of the association number in αB-crystallin oligomers in the model system, we referred to the AUC results.

The number of oligomers with an association number of na is defined as N[na].
2.2 We set n 1 and n 2 , which corresponded to the lower (= 20) and upper limit (= 32) of the association number of the αB-crystallin oligomer, respectively. To simplify the calculation, only the monomer was taken into consideration for the small molecular weight components.   3.6 To fix the ratio for k a to k d , we must consider the condition that r(cs) is equal to 0.015 regardless of cs.

Then, we set
The ratio of k a to k d was determined to 8.6 for both the dilute and the dense samples. This ratio was then utilized for the later calculation.
4. We set the parameters and a function as follows: • The rates of attaching and detaching monomers as k a and k d • Normalized forward scattering intensity at cs: I 0, nor (cs) = I 0 (cs)/I 0 (0) 5. Depending on the association number in αB-crystallin, three cases should be taken into consideration (Cases 5-1, 5-2, and 5-3). We then explain each case step by step.
The first case (Case 5-1) is that the association number of αB-crystallin is equal to n 2 . There exist two As a next step, we consider Case 5-2 in which the association number in αB-crystallin ranges from n 1 +1 to n 2 -1. There exist three cases that influence  the successive substitution method using Eqs. (S12), (S13) and (S14).
7. I 0 (cs) and I 0, nor (cs) are given by the following equation, respectively: